Practice Problems Test 3 Spring 2014
Use the information given about the angle θ, 0 ≤ θ ≤ 2π, to
find the exact value of the indicated trigonometric
function.
1
Find cos (2θ).
1) cos θ = - , csc θ < 0
7
14) sin2 θ + sin θ = 0
15) sin (4θ) = 16)
5
2) csc θ = - , tan θ > 0
2
Find cos (2θ).
4 3π
3) cos θ = , < θ < 2π
5
2
Find sin (2θ).
2 cos (2θ) = 1
17) 2 cos θ + 1 = 0
18) 2sin2 θ = 3(1 - cosθ)
19) cos(2θ) = 2 - 2sin2 θ
Use the given information to find the exact value of the
expression.
7
4) Find cos (2θ). sin θ = , θ lies in quadrant
25
20) 4(1 + sinθ) = cos2 θ
I.
21) 19x + 5cos x = 0
5) Find sin (2θ).
cos θ = 15
, θ lies in quadrant
17
22) x2 + 3sinx = 0
IV.
6) Find tan (2θ).
cot θ = **********************************************************
***NOTICE - For problems involving ʺSolve the Triangleʺ
the angles in this review are given by Greek letters:
A = α B = β C = γ
3
, θ lies in quadrant
3
III.
So, solution sets will look like this:
With alphabetical angels:
A = a =
B = b =
C = c =
Solve the equation. Give a general formula for all the
solutions.
7) sin θ = 1
8) sin θ = 3
2
3
2
9) cos (2θ) = With Greek lettered angles:
α = a =
β = b =
γ = c = *********************************************************
2
2
Use a calculator to solve the equation on the interval 0 ≤ θ
< 2π. Round the answer to two decimal places.
10) sin θ = -0.30
11) cos θ = 0.58
Solve the right triangle using the information given.
Round answers to two decimal places, if necessary.
23) b = 8, α = 40°; find a, c, and β
Solve the equation on the interval 0 ≤ θ < 2π.
12) 2 sin2 θ = sin θ
13) sec 2 θ - 2 = tan 2 θ
1
34) A guy wire to the top of a tower makes an
angle of 53° with the level ground. At a point
33 feet farther from the base of the tower and
in line with the base of the wire, the angle of
elevation to the top of the tower is 24°. What
is the length of the guy wire?
Solve the triangle. Round any answers to 2 decimal
places.
β = 50°,
a = 3
24) α = 30°,
For #ʹs 25 - 27, two sides and an angle are given.
Determine whether the given information results in one
triangle, two triangles, or no triangle at all. Solve any
triangle(s) that results. Round any answers to 2 decimal
places.
25) a = 7, b = 9, β = 49°
26) a = 8, b = 6, 27) a = 5,
b =69, α = 65°
Find the area of the triangle having the given
measurements. Round to the nearest square unit.
35) A = 27°
b = 11 in.
c = 7 in.
β = 15°
36) A = 35°
b = 17 m
c = 7 m
Two sides and an angle (SSA) of a triangle are given.
Determine whether the given measurements produce one
triangle, two triangles, or no triangle at all. Solve each
triangle that results. Round lengths to the nearest tenth
and angle measures to the nearest degree.
28) A = 30°
a = 8
b = 16
37) b = 24 in.
A = 18°
C = 72°
38) b = 10 ft
A = 18°
C = 96°
29) B = 102°
b = 3
a = 25
Solve the triangle. Round lengths to the nearest tenth and
angle measures to the nearest degree.
39) C = 112°
a = 8
b = 11
30) B = 19°
b = 8.9
a = 13.67
40) B = 63°
a = 12
c = 8
31) C = 35°
a = 18.7
c = 16.1
41) C = 120°
b = 5
a = 12
Solve the problem.
32) A surveyor standing 55 meters from the base
of a building measures the angle to the top of
the building and finds it to be 37°. The
surveyor then measures the angle to the top of
the radio tower on the building and finds that
it is 50°. How tall is the radio tower?
42) a = 7
b = 14
c = 16
Solve the problem.
43) Two sailboats leave a harbor in the Bahamas at
the same time. The first sails at 23 mph in a
direction 320°. The second sails at 31 mph in a
direction 190°. Assuming that both boats
maintain speed and heading, after 2 hours,
how far apart are the boats?
33) A ship sailing parallel to shore sights a
lighthouse at an angle of 15° from its direction
of travel. After traveling 3 miles farther, the
angle is 24°. At that time, how far is the ship
from the lighthouse?
2
44) Two points A and B are on opposite sides of a
building. A surveyor selects a third point C to
place a transit. Point C is 46 feet from point A
and 66 feet from point B. The angle ACB is 45°.
How far apart are points A and B?
50) (-2, 45°)
45) The distance from home plate to dead center
field in Sun Devil Stadium is 404 feet. A
baseball diamond is a square with a distance
from home plate to first base of 90 feet. How
far is it from first base to dead center field?
r
Use Heronʹs formula to find the area of the triangle.
Round to the nearest square unit.
46) a = 17 cm
b = 16 cm
c = 14 cm
51) (2, 45°)
47) a = 7 in
b = 11 in
c = 4 in
48) a = 11 in
b = 15 m
c = 7 m
r
Use a polar coordinate system to plot the point with the
given polar coordinates.
π
49) (2, - )
4
52) (-3, 690°)
r
r
3
53) (3, -150°)
56) (0, 2°)
r
54) (3, r
7π
)
6
57) (2, 360°)
r
r
Find another representation, (r, θ), for the point under the
given conditions.
π
58) 6, , r > 0 and 2π < θ < 4π
4
55) (2, 0°)
r
59) 3, π
, r < 0 and 0 < θ < 2π
2
60) 9, π
, r > 0 and -2π < θ < 0
6
61) 8, π
, r < 0 and 2π < θ < 4π
6
Polar coordinates of a point are given. Find the
rectangular coordinates of the point.
2π
62) 7, 3
4
63) 9, 3π
4
64) (-5, 120°)
65) (3, -135°)
66) (-2, 360°)
67) (-5, 36°)
The rectangular coordinates of a point are given. Find the
polar coordinates of the point.
68) (3, 0)
69) (0, -5)
70) (3, 3)
71) ( 3, 1)
72) (9, -9)
5
Answer Key
Testname: 113REVIEWT3SP14
1) - 2)
17
25
3) - 4)
47
49
24
25
527
625
5) - 240
289
6) - 3
π
7) {θ|θ = + 2kπ}
2
2π
π
+ 2kπ}
8) {θ|θ = + 2kπ, θ = 3
3
7π
π
+ kπ}
9) {θ|θ = + kπ, θ = 8
8
10) 3.45, 5.98
11) 0.95, 5.33
π 5π
12) 0, π, , 6 6
13) no solution
3π
14) 0, π, 2
15)
π π 2π 7π 7π 13π 5π 19π
, , , , , , , 12 6 3 12 6
12
3
12
16)
π 7π 9π 15π
, , , 8 8
8
8
17)
2π 4π
, 3
3
18) No Correct Answer Was Provided.
19) No Correct Answer Was Provided.
20) No Correct Answer Was Provided.
21) No Correct Answer Was Provided.
22) No Correct Answer Was Provided.
23) a = 6.71
c = 10.44
β = 50°
24) γ = 100°, b = 4.6, c = 5.91
25) one triangle
α = 35.94°, γ = 95.06°, c = 11.88
26) two triangles
α1 = 20.19°, γ1 = 144.81°, c1 = 13.36 or
α2 = 159.81°, γ2 = 5.19°, c2 = 2.1
27) one triangle
α = 35.94°, γ = 95.06°, c = 11.88
28) B = 90°, C = 60°, c = 13.9
6
Answer Key
Testname: 113REVIEWT3SP14
29) no triangle
30) A1 = 30°, C1 = 131°, c1 = 20.6;
A2 = 150°, C2 = 11°, c2 = 5.2
31) A1 = 42°, B1 = 103°, b1 = 27.4;
A2 = 138°, B2 = 7°, b2 = 3.4
32) 24.1 meters
33) 4.96 miles
34) 27.69 feet
35) 17 in.2
36) 34 m 2
37) 85 in.2
38) 17 ft2
39) c = 15.8, A = 28°, B = 40°
40) b = 11.0, A = 76°, C =41°
41) c = 15.1, A = 43°, B = 17°
42) A = 26°, B = 61°, C = 93°
43) 98.1 miles
44) 46.7 feet
45) 346.3 feet
46) 104 cm 2
47) 8 in 2
48) 34 m2
49)
5
4
3
2
1
-5
-4
-3
-2
-1
1
2
3
4
5 r
-1
-2
-3
-4
-5
7
Answer Key
Testname: 113REVIEWT3SP14
50)
5
4
3
2
1
-5
-4
-3
-2
-1
1
2
3
4
5 r
1
2
3
4
5 r
1
2
3
4
5 r
-1
-2
-3
-4
-5
51)
5
4
3
2
1
-5
-4
-3
-2
-1
-1
-2
-3
-4
-5
52)
5
4
3
2
1
-5
-4
-3
-2
-1
-1
-2
-3
-4
-5
8
Answer Key
Testname: 113REVIEWT3SP14
53)
5
4
3
2
1
-5
-4
-3
-2
-1
1
2
3
4
5 r
1
2
3
4
5 r
1
2
3
4
5 r
-1
-2
-3
-4
-5
54)
5
4
3
2
1
-5
-4
-3
-2
-1
-1
-2
-3
-4
-5
55)
5
4
3
2
1
-5
-4
-3
-2
-1
-1
-2
-3
-4
-5
9
Answer Key
Testname: 113REVIEWT3SP14
56)
5
4
3
2
1
-5
-4
-3
-2
-1
1
2
3
4
5 r
1
2
3
4
5 r
-1
-2
-3
-4
-5
57)
5
4
3
2
1
-5
-4
-3
-2
-1
-1
-2
-3
-4
-5
9
58) 6, π
4
3
59) -3, π
2
60) 9, - 61) -8, 11
π
6
19
π
6
7 7 3
62) - , 2
2
63)
-9 2 9 2
, 2
2
64)
5 -5 3
, 2
2
65)
-3 2 -3 2
, 2
2
66) (-2, 0)
67) (-4.045, -2.939)
10
Answer Key
Testname: 113REVIEWT3SP14
68) (3, 0)
π
69) (5, - )
2
70) (3 2, π
)
4
π
71) (2, )
6
72) (9 2, 7π
)
4
11